Problem: What is the average rate of change of the function $g(x)=\ln(x)$ over the interval $[7,t]$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\ln(t-7)}{t}$ (Choice B) B $\dfrac{\ln(t)-\ln(7)}{t-7}$ (Choice C) C $\dfrac{\ln(t-7)}{t-7}$ (Choice D) D $\dfrac{\ln(t)-\ln(7)}{7}$
Explanation: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We are interested in the average rate of change of $g(x)=\ln(x)$ over the interval $[7,t]$ : $\begin{aligned} &\phantom{=}\dfrac{g(t)-g(7)}{(t)-(7)} \\\\ &=\dfrac{\ln(t)-\ln(7)}{t-7} \end{aligned}$ The average rate of change of the function is $\dfrac{\ln(t)-\ln(7)}{t-7}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints.